Rotating eigenstates of J operator into each other?

where
[tex]
\hbar^2 j(j+1) , \hbar m
[/tex]
are the eigenvalues of J^2 and Jz, respectively. Is it always possible to rotate these states into each other? i.e. given |j,m> and |j,m'>, is it always possible to find a unitary rotation operator U^j such that
[tex]
|j,m' > = U^{(j)} |j, m > [/tex]

I think the above would work. But do you consider that a rotation? I suppose one might consider all special unitary operators to be rotations in a sense (rotations in the Hilbert space). However I don't know if you mean rotations as in rotations in 3D space, that is ## U = e^{-i\theta \hat{\mathbf{n}} \cdot \mathbf{J}} ##. If this is what you mean, then I'm not sure of the answer. I know such rotations will often result in superpositions of m states, which is not what you want.